chapter 6 river water quality modeling
TRANSCRIPT
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Chapter 6
River Water Quality Modeling
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Chapter 6 River Water Quality Modeling
Contents
6.1 Non-Conservative Pollutants
6.2 Modeling BOD-DO Coupled System
6.3 Modeling Heat Transport
6.4 Eutrophication
Objectives
- Classify non-conservative pollutants
- Present concept of water quality modeling
- Study BOD-DO coupled system in the river
- Study transport of heat and algae
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6.1 Non-Conservative Pollutants
6.1.1 Category of Non-Conservative Pollutants
1) Toxic Substance
- Metals: mercury, cadmium, lead
- Industrial chemicals: toluene, benzenes, phenols, PCB
- Hydrocarbons: PAH (polycyclic aromatic hydrocarbons)
- Agricultural chemicals: pesticides, herbicides, DDT
- Radioactive substances
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6.1 Non-Conservative Pollutants
2) BOD-DO coupled system
3) Temperature
4) Nutrients and eutrophication
5) Bacteria and pathogens
6) Oil
[Cf] Conservative pollutants
- one which does not undergo any chemical or biochemical changes in
transport
- no loss due to chemical reactions or biochemical degradation
- salt, chloride, total dissolved solids, some metals
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6.1 Non-Conservative Pollutants
6.1.2 Transport of Non-Conservative Pollutants
(1) Toxic Substance
Physio-chemical phases of the transport of toxic substances:
- loss of the chemical due to biodegradation, volatilization, photolysis,
and other chemical and bio-chemical reactions
- sorption and desorption between dissolved and particulate forms in
the water column and bed sediment
- settling and resuspension mechanisms of particulates between water
column and bed sediment
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6.1 Non-Conservative Pollutants
Assume only loss of the chemical
( ) ( ) ( ) ( )hC uCh vCh hD C hSt x y
∂ ∂ ∂+ + =∇ ⋅ ∇ +
∂ ∂ ∂
where S = sink/source term
Assume first-order decay
- decay rate is proportional to the amount of material present
dC kC Sdt
= − =
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6.1 Non-Conservative Pollutants
C S kwhere = mass/volume; = mass/(volume‧time); = 1/time = decay rate
- Rate of disappearance of BOD due to biodegradation
- Radioactive substance also decay in strength in this way
- Coliform bacteria and pathogens die away with a rate of first-order decay
( ) ( ) ( ) ( )hC uCh vCh hD C khCt x y
∂ ∂ ∂+ + =∇ ⋅ ∇ −
∂ ∂ ∂
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6.1 Non-Conservative Pollutants
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6.1 Non-Conservative Pollutants
(2) BOD-DO
- Linked materials
- Behavior of one material depends upon the amount of another- Conc.
of dissolved oxygen depends not only on transport of DO but also on
the amount of BOD present
- Biodegradable substances undergo biochemical reactions
- Oxygen is used up in aerobic decomposition
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6.1 Non-Conservative Pollutants
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6.1 Non-Conservative Pollutants
(3) Heat transport
Heat and temperature
- Heat is the extensive quantity whereas temperature is intensive
(ex. Mass is the extensive property whereas concentration is intensive
(size-independent))
- Discharge of excess heat from industrial or municipal effluents may
positively or negatively affect the aquatic ecosystem
- Strong influence many physiological and biochemical processes
- Control of the rate of biological and chemical reactions
- Oxygen solubility governed by water temperature (the colder the water,
the more the dissolved oxygen)
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6.1 Non-Conservative Pollutants
The heat exchange with the sediment bed is generally much smaller than
the surface exchange and is frequently neglected in modeling studies
(Morin and Couillard (1990), Hondzo and Stefan (1994), Younus et al.
(2000)).
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6.1 Non-Conservative Pollutants
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6.1 Non-Conservative Pollutants
- Eutrophication is excessive nutrients such as nitrogen and phosphorus
in ecosystems.
- One example is algal bloom by great increase of algae.
- Adverse environmental effects involve the depletion of oxygen in the
water bodies which cause a reduction in aquatic animals with water
quality deterioration.
- Nitrogen and phosphorus are crucial indicators to assess
eutrophication level.
- Algae and nutrients are transported by advection and dispersion with
complex physicochemical processes.
(4) Eutrophication
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6.1 Non-Conservative Pollutants
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6.1 Non-Conservative Pollutants
(5) Bacteria and Pathogens
- Related to waterborne diseases (e.g., gastroenteritis, amoebic
dysentery, cholera, etc.)
- The modes of transmission of pathogens are through drinking water,
primary & secondary contact recreation, etc.
- Examples of communicable disease indicators and pathogens
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6.1 Non-Conservative Pollutants
Type Organisms
Indicator bacteriaTotal Coliform, Fecal Coliform, E. Coli, Fecal streptococci,
Enterococci, etc.
PathogensVibrio cholera, Salmonella species, Shigella species,
Giardia lambia, Entamoeba histolytica, etc.
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6.1 Non-Conservative Pollutants
- The principal sources of organisms:
(a) point sources from domestic, municipal, and some industrial sources
(b) combined sewer overflows
(c) runoff from urban and suburban land
(d) municipal waste sludges disposed of on land or in water bodies
- Decay rate
1B B BI Bs aK K K K K= + + −
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6.1 Non-Conservative Pollutants
where, = basic death rate as a function of temperature, salinity,
predation,1BK
BIK
BsKaK= death rate due to sunlight, = net loss due to settling (resuspension)
= after growth rate
- For rivers and streams, the downstream distribution of bacteria is
0 exp( )BN N K t∗= −
0N
BK /t x U∗ =
where, = the concentration at the outfall after mixing [num/L3],
=the overall net first-order decay rate [1/day],
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6.1 Non-Conservative Pollutants
(6) Oil
i) Advection
- Advection of oil is recognized as a three-dimensional process, with
key mechanisms occurring over a wide range of scales
- Oil moves horizontally in the water under forcing from wind, waves
and currents
- Transported vertically in the water column in the form of droplets of
various sizes
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6.1 Non-Conservative Pollutants
ⅱ) Spreading
- Oil film thickness determines the persistence of the oil on the water
surface
- Oil slick area (film thickness) is used in the computation of
evaporation, which determines changes in oil composition and
properties with time
- For instantaneous spills, Fay-type spreading model (Fay, 1971)
provides adequate predictions of the film thickness
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6.1 Non-Conservative Pollutants
1/82 6
2 3 6~ VAD s
σρ ν
σ Vρ ν
D s
where = spreading coefficient or interfacial tension, = volume of oil in
axisymmetric spread, = density of water, = kinematic viscosity of water,
= diffusivity of the surfactants in water, = solubility
ⅲ) Evaporation
- Estimates of evaporative losses are required in order to assess the
persistence of the spill, and are also the basis for estimates of changes
in oil properties with time
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6.1 Non-Conservative Pollutants
- During a spill, approximately 25 ~ 40% of the total mass can be lost
by evaporation alone, depending on the environmental conditions
and the type of oil (Azevedo et al., 2014)
- Evaporative exposure formulation (Stiver and Mackay, 1984)
( )00
10.3exp 6.3v eG v
dF K A T T Fdt V T
= − +
vF t 3 0.782.0 10e wK U−= × × wU
A T K 0T
GT
where = fraction evaporated, = time, , = wind
velocity, = film thickness, = environmental temperature ( ), and
= oil-dependent parameters derived from the fractional distillation data
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6.1 Non-Conservative Pollutants
ⅳ) Natural dispersion
- Computation of natural dispersion is required for assessment of lifetime
of an oil spill
- The rate of natural dispersion depends on environmental parameters,
but is also influenced by oil-related parameters (oil film thickness,
density, surface tension and viscosity)
- Experimental work of Delvigne and Sweeny (1988) revealed that the
number of droplets in a certain diameter class could be related to the
droplet size with a common power law relationship, independent of the
type of oil and the wave conditions
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6.1 Non-Conservative Pollutants
1.4 1.7d DQ aH D≤ =
d DQ ≤
,D a
H
where = entrained oil mass per unit area included in droplets up to
a certain diameter = dispersion coefficient which is related to the
oil type in terms of the oil viscosity, = breaking wave height
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6.1 Non-Conservative Pollutants
Photolysis
Drifting
Spreading(by gravity, inertia, viscous,
and interfacial tension)
Oil slick
Evaporation
Dispersion in water(by small-scale turbulent eddies)
Dissolution of dispersed oil
Sinking
RiseCoalesce
Suspended solid
Sorption
Emulsification
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6.2 Modeling BOD-DO Coupled System
6.2.1 Solutions of BOD-DO Coupled System
◆ Coupled system (BOD, DO)
- determination of dissolved oxygen concentrations downstream of a
discharge of BOD
◆ Oxygen Demand
= indirect measure of organic materials (= organic pollutants) in terms of
the amount of oxygen required to (completely) oxidize it.
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6.2 Modeling BOD-DO Coupled System
COD - Chemical Oxygen Demand
BOD: CBOD - Carbonaceous BOD
NBOD - Nitrogeneous BOD
Organic matter + 2 2 2O CO H O→ +
◆ Importance of dissolved oxygen (DO)
- Anaerobic conditions in a stream are indicative of extreme pollution
- Low dissolved oxygen concentrations have severe effects on the kind of
biota which inhabit the stream
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6.2 Modeling BOD-DO Coupled System
◆ Sources and sinks of DO
(a) Sources
- Reaeration from the atmosphere
- Photosynthetic oxygen production
- DO in incoming tributaries
(b) Sinks
- Oxidation of BOD
- Oxygen demand of sediments of water body
- Use of oxygen for respiration by aquatic plants
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6.2 Modeling BOD-DO Coupled System
dCdt
∴ = reaeration + (Photosynthesis-respiration) − oxydation of BOD
− sediment oxygen demand ± oxygen transport (into and out of segment)
Let C = concentration of DO
L = concentration of BOD
1dL k Ldt
= −
1k
(1) rate of utilization of DO by BOD
→ exertion of BOD
= utilization of DO
= depletion of DO
= deoxygenation coefficient
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6.2 Modeling BOD-DO Coupled System
sC
sC C−
(2) reaeration from the atmosphere
= diffuse of oxygen into the stream rate of reaeration
degree to which the water is unsaturated with oxygen
Let = DO saturation concentration
then oxygen deficit, DOD =
∴ rate of reaeration
∝
2 ( )sdC k C Cdt
= + −
2k = reaeration coefficient (1/T)
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6.2 Modeling BOD-DO Coupled System
∴ Conservation equation for C2
1 22 ( )sC C Cu E k L k C Ct x x
∂ ∂ ∂= − + − + −
∂ ∂ ∂
sD C C= −
dD dC= −
Let
Then
2
1 22D D Du E k L k Dt x x
∂ ∂ ∂∴− = + − − +
∂ ∂ ∂2
1 22D D Du E k L k Dt x x
∂ ∂ ∂= − + + −
∂ ∂ ∂⇒ G.E. for DO Deficit
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6.2 Modeling BOD-DO Coupled System
2
12
L L Lu E k Lt x x
∂ ∂ ∂= − + −
∂ ∂ ∂
L rS k L= −
D d aS k L k D= −
For BOD
⇒ G.E. for BOD concentration
Let reaction terms
in whichL
D sC C−
= concentration of remaining BOD
= dissolved oxygen deficit (DOD) =
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6.2 Modeling BOD-DO Coupled System
sC
C
rkd sk k+
sk
dk
ak
= DO saturation concentration
= actual DO concentraion
= BOD removal coefficient =
= settling coefficient
= deoxygenation coefficient = biochemical degradation
= reaeration coefficient
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6.2 Modeling BOD-DO Coupled System
2
2 rL L Lu E k Lt x x
∂ ∂ ∂= − + −
∂ ∂ ∂2
2 d aD D Du E k L k Dt x x
∂ ∂ ∂= − + + −
∂ ∂ ∂
G.E.: unsteady state
DOD
BOD
◆ Steady state, W/ Dispersion (Estuary)2
20 rL Lu E k Lx x∂ ∂
= − + −∂ ∂
(i) BOD: → same as case 4
( )0 exp 1 , 02 ruL L x xE
α = + ≤
( )0 exp 1 , 02 ruL x xE
α = − ≤
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6.2 Modeling BOD-DO Coupled System
in which0
r
WLQα
=
241 r
rk Eu
α = +
(ii) DO: 2
20 d aD Du E k L k Dx x
∂ ∂= − + + −
∂ ∂
( )exp (1 ) exp 12 2 , 0
r ad
a r r a
u ux xW k E ED xQ k k
α α
α α
+ − = − ≤ −
( )exp (1 ) exp 12 2 , 0
r ad
a r r a
u ux xW k E ED xQ k k
α α
α α
− + = − ≥ −
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6.2 Modeling BOD-DO Coupled System
in which
241 a
ak Eu
α = +
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6.2 Modeling BOD-DO Coupled System
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6.2 Modeling BOD-DO Coupled System
0E =
6.2.2 Streeter-Phelps Equation
◆ Streeter-Phelps Equation (1925)
- no dispersion (river)
- steady state
10 Lu k Lx∂
= − −∂
1 20 Du k L k Dx
∂= − + −
∂
BOD:
DO:
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6.2 Modeling BOD-DO Coupled System
For BOD we have solution (Case 3)
1 10 exp( ) exp( )k W kL L x x
u Q u∴ = − = −
B.C.: 0 0(0) sD D C C= = −
Solution:
( )1 2 2
10 0
2 1
, 0k k kx x xu u ukD x L e e D e x
k k
− − −
= − + ≥
− (1)
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6.2 Modeling BOD-DO Coupled System
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6.2 Modeling BOD-DO Coupled System
cD ct◆ Critical deficit (@ uptake of oxygen by BOD is just balanced by
the input of oxygen from atmosphere)
let x/u =t (= time of flow) time of travel
→ Then Eq. (1) becomes
( ) 1 2 210 0
2 1
, 0k t k t k tkD t L e e D e xk k
− − − = − + ≥ −(2)
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6.2 Modeling BOD-DO Coupled System
ct→ may be found as
0;Dt
∂=
∂( )2 12 0
2 1 1 1 0
1 ln 1ck kk Dt
k k k k L − = − −
110
2
ck tc
kD L ek
−= ← from Eq.(2)
◆ Modified Streeter-Phelps equation
- account for other processes
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6.2 Modeling BOD-DO Coupled System
(1) BOD removal due to sedimentation
- non-oxygen-demanding
- reduces BOD w/o changing oxygen concentration
3L k Lt
∂= −
∂
A
(2) BOD addition due to scour from the bottom and surface runoff from the
land
- rate of addition is assumed to be constant
(3) Oxygen use other than by aerobic biochemical oxygen demand in the
water, and oxygen addition other than through reaeration.
- net of these processes is constant rate
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6.2 Modeling BOD-DO Coupled System
◆ Solution :
( )( ){ }1 2 21
02 1 3 1 2
k k t k tt
k PD L e ek k k k k
− + − = − − − + +
( )21
2 1 2 1
1 k tk P A ek k k k
− + − − +
3k A PConsider only neglect &
1 3,rk k k= + 1,dk k= 3,sk k= 2ak k=
{ }0 0a ar k t k tk td
ta r
kD L e e D ek k
− −−= − +−
Let
Then
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6.2 Modeling BOD-DO Coupled System
[Remark]
0 rLu k Lx∂
= − −∂
0 d aDu k L k Dx
∂= − + −
∂
G.E.: BOD
DOD
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6.3 Modeling Heat Transport
(a) Sources
- Shortwave solar radiation
- Longwave atmospheric radiation
- Conduction of heat from atmosphere to water
- Direct heat input from municipal and industrial activities
(b) Sinks (losses)
- Longwave radiation emitted by water
- Evaporation
- Conduction from water to atmosphere
◆ Sources and sinks of heat
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6.3 Modeling Heat Transport
◆ Heat balance equation
Edinger and Geyer (1965) and Edinger et al. (1974) provide a review of
each of the process.
net s a b c eq q q q q q= + + + +
netq
sq
aq
bq
cq
eq
where
= net heat exchange across the water surface
= shortwave solar radiation
= longwave atmospheric radiation
= longwave radiation from water
= conductive heat transfer
= evaporative heat transfer
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6.3 Modeling Heat Transport
2cal/cm day⋅All terms are in units such as.
◆ Simplified heat balance equation
Edinger et al. (1974) have shown that the net heat input can be represented by
( )net eq K T T= −
K 2W/m C°
eT
where
= surface heat exchange coefficient ( )
= equilibrium temperature
= temperature that a body of water would reach if all
meteorological conditions were constant in time
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6.3 Modeling Heat Transport
K
4.5 0.05 ( ) 0.47 ( )w wK T f U f Uβ= + + +
◆ Exchange coefficient,
Edinger et al. (1974) proposed as follows,
( )wf U 29.2 0.46 wU= + 2(W/m mm Hg)⋅
wU
β 20.35 0.015 0.0012m mT T= + +
( ) / 2m dT T T= +
dT
where
= wind function
= dew point temperature
= wind speed in m/s (measured at a height of 7 m above the water
surface)
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6.3 Modeling Heat Transport
eT◆ Equilibrium temperature,
The equilibrium temperature can be estimated for a given set of
meteorological conditions by iteration until qnet = 0. Alternately, under
constant coefficients, the equilibrium temperature as given by Edinger et
al. (1974), is approximated by the empirical relationship
se d
qT TK
= +
◆ Time rate of change of temperature
( )net e
p p
dT q K T Tdt c h c hρ ρ
−= =
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6.3 Modeling Heat Transport
ρ 3(g/cm )
pc (1 cal/g C)°
where
= water density
= specific heat of water
( ) ( ) ( )hT uTh vTh hD T hSt x y
∂ ∂ ∂+ + = ∇⋅ ∇ +
∂ ∂ ∂
( )net e
p p
dT q K T TSdt c h c hρ ρ
−= = =
◆ Heat transport equation
, ,u v hAssume that satisfy the continuity eq.
1 ( ) ( )ep
T T T Ku v hD T T Tt x y h cρ
∂ ∂ ∂+ + = ∇ ⋅ ∇ + −
∂ ∂ ∂
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6.4 Eutrophication
6.4.1 Modeling Nitrogen and Phosphorus
◆ Transport of nitrogen and phosphorus
- Advection and dispersion are the most important mechanisms in the
nutrient transport
- Nutrients involve chemical reactions or biological evolutions
- General partial differential equation of nutrients for a one-dimensional
model:2
2∂ ∂ ∂
= − + ± −∂ ∂ ∂C C Cu E S kCt x x
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6.4 Eutrophication
C
E
S
k
where
= concentration of nutrients
= longitudinal dispersion coefficient
= sink and source by external contribution
= first-order decay
( , )± − =S kC R C t2
2 ( , )C C Cu E R C tt x x
∂ ∂ ∂= − +
∂∴ +
∂ ∂
Let
where
( , )R C t = reaction term of nutrients
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6.4 Eutrophication
◆ Nitrogen cycle
- The primary source of nitrogen is agricultural soil management with
synthetic fertilizers, accounting for about 74% of total NO2 emission in
2013 (USEPA, 2015).
- The nitrogen cycle considers ammonia nitrogen (NH4-N), nitrite nitrogen
(NO2-N), nitrate nitrogen (NO3-N), and organic nitrogen (Org-N).
- Nitrification and denitrification are important phases in the nitrogen cycle
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6.4 Eutrophication
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6.4 Eutrophication
2 4 3- 3 2 + − −+ → + +Org N H O NH HCO OH
◆ Reaction terms of nitrogen
(1) organic nitrogen (Org-N)
- Source: respiration by algae
- Decay: ammonification from organic nitrogen to ammonia nitrogen, and
settling
- Ammonification:
( , )∴ orgR N t
,( 20) ( 20), , , , n orgT T
n A r A A n org n org orgk A k Nh
ωα θ θ− −
= − +
= respiration – ammonification – settling
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6.4 Eutrophication
orgN
,αn A
,r Ak
Aθ
,n orgθ
T
A
,n orgk
,ωn org
h
where
= concentration of organic nitrogen
= nitrogen content in algae
= algal respiration rate
= temperature coefficient for algae
= temperature coefficient for organic nitrogen
= water temperature
= concentration of algae
= rate of ammonification of organic nitrogen into ammonia nitrogen
= rate of organic nitrogen settling
= water depth
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6.4 Eutrophication
4 2 2 21.5 2+ − ++ → + +NH O NO H O H
(2) ammonia-nitrogen (NH4-N)
- Source: ammonification from organic nitrogen to ammonia-nitrogen
- Decay: nitrification from ammonia-nitrogen to nitrite-nitrogen
- Nitrification (a):
( , )∴ ammR N t
( 20) ( 20), , , , T T
n org n org org n amm n amm ammk N k Nθ θ− −= −
= ammonification – nitrification (a)
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6.4 Eutrophication
ammN
,n ammk
,n ammθ
where
= concentration of ammonia-nitrogen
= nitrification rate of ammonia-nitrogen into nitrite-nitrogen
= temperature coefficient for ammonia-nitrogen
(3) nitrite-nitrogen (NO2-N)
- Source: nitrification from ammonia-nitrogen to nitrite-nitrogen
- Decay: nitrification from nitrite-nitrogen to nitrate-nitrogen
Nitrification (b): 2 2 30.5− −+ →NO O NO
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6.4 Eutrophication
( , )∴ nitriR N t = nitrification (a) – nitrification (b)
( 20) ( 20), , , , T T
n amm n amm amm n nitri n nitri nitrik N k Nθ θ− −= −
nitriN
,n nitrik
,n nitriθ
where
= concentration of nitrite-nitrogen
= nitrification rate of nitrite-nitrogen into nitrate-nitrogen
= temperature coefficient for nitrite-nitrogen
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6.4 Eutrophication
(4) nitrate-nitrogen (NO3-N)
- Source: nitrification from nitrite-nitrogen to nitrate-nitrogen
- Sink: uptake by algae
- Decay: denitrification from nitrate-nitrogen to nitrogen gas (N2)
Denitrification: 2 3 2 2 21.25 0.5 1.25 1.75− ++ + → + +CH O NO H N CO H O
( , )∴ nitraR N t
( 20) ( 20), , , , , T T
n nitri n nitri nitri n nitra n nitra nitra n Ak N k N Aθ θ α µ− −= − −
= nitrification (b) – denitrification – uptake
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6.4 Eutrophication
nitraN
,n nitrak
,n nitraθ
µ
where
= concentration of nitrate-nitrogen
= denitrification rate of nitrate-nitrogen into nitrogen gas
= temperature coefficient for nitrate-nitrogen
= algal growth rate
◆ Phosphorus Cycle
- The phosphorus cycle has no major gaseous component.
- Phosphorus loading contributed by runoff from pastures and croplands
with livestock waste and fertilizers (USGS, 2000)
- The phosphorus cycle includes organic phosphorus (Org-P), and
dissolved phosphorus or phosphate phosphorus (PO4-P).
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6.4 Eutrophication
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6.4 Eutrophication
◆ Reaction terms of phosphorus
(1) organic phosphorus (Org-P)
- Source: respiration by algae
- Decay: mineralization from organic phosphorus to phosphate-
phosphorus, and settling
( , )∴ orgR P t
,( 20) ( 20), , , , p orgT T
p A r A A p org p org orgk A k Ph
ωα θ θ− −
= − +
-
= respiration – mineralization – settling
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6.4 Eutrophication
orgP
,α p A
,p orgk
,p orgθ
,ωp org
where
= concentration of organic phosphorus
= phosphorus content in algae
= mineralization of organic phosphorus into phosphate phosphorus
= temperature coefficient for organic phosphorus
= rate of organic phosphorus settling
(2) phosphate-phosphorus (PO4-P)
- Source: mineralization, excretion from algae, and aerobic release
from sediment
- Sink: uptake by algae
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6.4 Eutrophication
( , )∴ dissR P t
( ),( 20) ( 20), , , , p dissT T
p org p org org p A e A Ak P k Ah
γθ α θ µ− −= + + −
= mineralization + excretion + release – uptake
dissP
,γ p diss
,e Ak
where
= concentration of phosphate phosphorus
= rate of aerobic release from sediment
= algal excretion rate
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6.4 Eutrophication
6.4.2 Modeling Algae
◆ Characteristics of algae
- Aquatic photosynthesis micro-organisms
- Diatom, green algae and cyanobacteria are common species in the
water systems
- The presence of algae in the river depends on the factors including:
nutrients, temperature, and sunlight intensity (Hornbeger and Kelly,
1975; Zhen-Gang, 2008).
- Coupled with nitrogen and phosphorus cycle
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6.4 Eutrophication
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6.4 Eutrophication
◆ Transport of algae
- Similar to the nutrient transport
- Growth rate is added instead of sink-source in the reaction term.
- General partial differential equation of algae for a one-dimensional model:2
2 µ∂ ∂ ∂= − + + −
∂ ∂ ∂A A Au E A kAt x x
( , )µ − =A kA R A t2
2 ( , )A A Au E R A tt x x
∂ ∂ ∂+
∂∴ = − +
∂ ∂
Let
( , )R A twhere
= reaction term of algae
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6.4 Eutrophication
◆ Reaction term of algae
- Growth: photosynthesis (or uptake of nitrate nitrogen and phosphate
phosphorus)
- Algal growth is a function of temperature, light, and nutrients (Bowie, 1985)
- Decay: respiration, excretion, grazing by zooplankton, and settling
Photosynthesis: 22 3 4 2 106 236 110 16 2106 16 122 138− −+ + + → +CO NO HPO H O C H O N P O
( 20) ( 20) ( 20)max , , ,( , ) ( ) ( ) ( ) T T T A
r A A e A A z A AR A t f T f N f I k k k Ahωµ θ θ θ− − − = ⋅ ⋅ ⋅ − − − −
∴
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6.4 Eutrophication
maxµ
( )f T
( )f N
( )f I
zk
ωA
where
= maximum growth rate at a particular reference temperature
= Temperature limiting factor on local growth rate
= Nutrient limiting factor on local growth rate
= Light limiting factor on local growth rate
= grazing rate by zooplankton
= rate of algal settling
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(1) Temperature limitation
- Three major categories of a temperature adjustment function are used
to model algae:
a) Linear function (Bierman et al., 1980; Canale et al., 1975):
minopt
opt min opt min
opt
1( ) for
( ) 1 for
Tf T T T TT T T T
f T T T
= − ≤ − −
= >
where
= optimal temperature for algal growth
= minimum temperature for algal growthoptT
minT
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b) Exponential function (Eppley, 1972):
( )( ) refT Tf T θ −=
refTwhere
= reference temperature for algal growth ( = 20˚C)
c) Skewed normal distribution function (Cerco and Cole, 1995):
21 opt opt
22 opt
exp - ( - ) , if ( )
exp - ( - ) , otherwise
KTg T T T Tf T
KTg T T
≤ =
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1KTg
2KTg
where
= rate coefficient for left side of curve
= rate coefficient for right side of curve
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(2) Nutrient limitation
- Monod model (1945) is frequently used for considering effect of limiting
nutrients as substrates on the growth of micro-organisms.
Nφ =+S
SK S
S
SK
where
= concentration of the limiting nutrient
= half-saturation constant of the limiting nutrient
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- Three primary approaches are used to determine the combined
effect of the nutrients:
a) Multiplicative:
Nφ = ⋅+ +
nitra diss
n nitra p diss
N PK N K P
b) Limiting nutrient (or Liebig’s minimum law):
N min , φ
= + + nitra diss
n nitra p diss
N PK N K P
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c) Harmonic mean:
N 2φ
++ +
=
nitra diss
n nitra p diss
N PK N K P
nK
pK
where
= half-saturation rate of nitrate nitrogen
= half-saturation rate of phosphate phosphorus
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(3) Light limitation
- Three formulas are generally used to estimate the light effect on algal
growth rate:
a) Michaelis-Menten (saturation) model:
φ =+L
si
IK I
b) Steele (photoinhibition) model (1962):1
φ−
= ⋅ S
II
LS
I eI
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c) Smith (hyperbolic saturation) model (1936):
2 2φ =
+L
k
II I
siK
SI
kI
where
= half-saturation constant of sunlight intensity
= Steele’s constant
= Smith’s constant
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